Organizers had already won the right to hold a vote at a different facility in Staten Island, scheduled for between March 25 and 30. To move forward with an election, union organizers typically have to prove they have gained signatures from at least 30% of workers at a facility.

“Two groups of Amazon workers in NYC are set to make their voice heard in the coming months, so please keep updated & support our grassroots campaign in any way possible!” the group of independent workers who have organized in Staten Island said in a tweet late Wednesday.

An Amazon spokeswoman said the company looks forward to having its employees’ voices heard, and that its focus “remains on working directly with our team to make Amazon a great place to work.” LDJ5 has roughly 1,500 employees, according to organizers.

The date and conditions for the vote still need to be determined.

No Amazon employees in the U.S. are currently represented by a union, but organizers have been pressing their cause in several locations.

A group of organizers who call themselves Amazon Labor Union have campaigned for months in Staten Island, saying they seek changes to pay and working conditions. They say the company requires workers to package and sort items at a rapid pace, sometimes hundreds of items an hour.

Amazon says that it offers relatively high pay for warehouse positions, with wages averaging about $18 an hour, as well as benefits including healthcare and 401(k) options.

Staten Island is home to four Amazon facilities, the largest being JFK8, where organizers and the company agreed last month to move forward with an election.

Chris Smalls,

a former Amazon employee leading the Staten Island group, said he aims to unionize all four.

Amazon faces a repeat election at a warehouse in Bessemer, Ala., where workers are voting by mail and have until the end of the month to submit their ballots. The vote is being held again after the NLRB found Amazon violated labor law during the first election a year ago, when workers voted against unionizing. Amazon contested the charge.

The company has said it opposes unions because it prefers to negotiate with workers directly. It has held meetings at its sites to discourage workers from supporting the union, according to workers who say the company often points out potential union fees workers could have to pay. High rates of employee turnover at many Amazon facilities also make it difficult for organizers to establish and maintain support.

Write to Sebastian Herrera at Sebastian.Herrera@wsj.com